APPLICATION OF FUZZY CLUSTERING TO VIBRATION FAULT DIAGNOSIS OF HYDROELECTRIC GENERATING SET

Chen Qijuan    Chen Tiehua

Wuhan University

Address: 8 Donghu Nanlu, Wuhan  430072  China

Tel: +86-27-67802270, Fax: +86-27-67802272, E-mail:qjchen@wuhee.edu.cn

Abstract: The implementation of the hydroelectric equipment condition monitoring and diagnosing system is gradually processing from the theory study towards practical exploitation, and into a further practice in hydropower stations. Generally this work is beginning with vibration monitoring and analysis of hydroelectric generating set (sometimes called unit). However, the unit has a combined influence from water, machine, electricity and many other factors. The causes resulted in vibration are complicated. Here, the membership concept of fuzzy mathematics is employed to identify possible existence of specific faults linked to vibration characteristics of hydroelectric generating set. Dynamic fuzzy clustering is applied to analyze the vibration causes. The fuzzy dynamic ISODATA(Iterative self－Organizing Data Analysis Technique  A) algorithm is also introduced to diagnose the unit vibration faults. It is feasible and has a degree of reliability. In which a fuzzy relationship matrix, corresponding symptom matrix and related fault matrix can be constructed based on each unit’s structure and design parameters. They are realized by changing the columns and row of matrix. It shows by a real example that calculation results are in good agreement with actual inspection records. It is considered that this method has good prospects for future use. 

Keywords: hydroelectric generating set, vibration, diagnosis, membership, fuzzy clustering

1    INTRODUCTION

Regulating frequency or peak load within electric power supply systems is usually undertaken by large-scale hydroelectric generating set, its safe and stable operation is very important. Vibration is one of the most important features indicating stability of a unit’s operation. So far, vibration diagnosis has been the basic method of condition monitoring and fault diagnosis. The complexity and combined effects of hydraulic, mechanical and electrical factors create non-unique relationships between specific faults and the corresponding vibration symptoms of a hydroelectric generating set. It is urgently required to Research relationships between complex vibration faults and vibration signals recorded during condition monitoring and fault diagnosis. The method of fuzzy clustering analysis is proposed to do this.

2    THEORY OF FUZZY CLUSTERING ANALYSIS

2.1    Samples and construction of membership function

Samples are vibration features, including symptoms of vibration faults. The Cauchy half ascendancy distribution function is generally employed to determine membership function of fault symptoms in fuzzy diagnosis ^[1]. This is shown as follows:

                 (1)

Where ‘a’ is always equal to zero, while ‘k’ depends on the characteristics of different units. Taking criteria for ‘heavy vibration’ as an example, there are standard limiting vibration values for each part of a hydroelectric generating set. When its amplitude is equal to the limiting value, let the corresponding membership grade of ‘heavy vibration’ be 0.5. The value of ‘k’ can be obtained from equation (1), by setting µ_x and x, respectively, equal to 0.5 and the value of limiting amplitude.  Limiting amplitudes are different at different rated rotating speeds. So ‘k’ should be calculated in this way for the different limiting amplitudes.

2.2    Construction of fuzzy relationship function

From fuzzy theory, fuzzy relationship function can be constructed as [u_mn]=R [u_ln], where [u_ln] is the matrix of symptom membership grade; [u_mn] is the matrix of fault membership grade; R is the fuzzy relationship matrix between fault and symptom. R is usually developed from expert experience.

2.3    Method of dynamic fuzzy isodata analysis ^[2,3]

Consider an object set X={x₁, x₂, …x_n}, in which every sample x_i has m feature norms, namely, x_i = {x_i1, x_i2, …, x_im}. All the samples are classified into C clusters(2≤C≤N).

A cluster criterion is necessary to select the optimal fuzzy partition from the classification space. So an Objective Function is defined as:

                       (2)

where R = (r_ik)_c×n  is the fuzzy partition matrix, r_ik∈[0,1]; v_i is the cluster center, where a cluster is located and around which its objects are concentrated; q is an exponential weight. The greater q exceeds 1, the fuzzier the final partition becomes. J(R,V) represents the sum of squared distances from each sample to its cluster center. ‖x_k-v_i‖represents the distance from the sample x_k to the cluster center of cluster i. Here Euclidian distance is employed:

                        (3)

Typically, the local extreme of an objective function is defined by an optimal clustering criterion, namely a minimum value of {J(R ,V)}. When J(R ,V) reaches a minimum, the corresponding values of r_ik and v_i are:

                (4)                       (5)

It is very difficult to obtain the minimum of an objective function in fuzzy clustering. Fortunately, Bezdek proved that^[3], when q≥1 and x_k≠v_i, the ISODATA algorithm ensures convergence of a calculation. So an optimal partition can be obtained using ISODATA, after ideal samples for clustering are calculated as given above. Detailed steps are as follows:

(1) Choose the initial number of clusters C; choose an initial partition  (that is, all the samples are classified artificially. Crisp (or hard) clustering can be applied).

(2) Calculate the cluster central vectors V_i by using equation (5).

(3) Calculate a new membership matrix R based on V_i and equation (4).

(4) If max(|r_ik^*-r_ik| )<ε, then stop. R^* and V are identified, otherwise return to step 2, repeating the above calculation steps.

εis a pre-selected small positive value. The smaller the value of εis, the more accurate the results will be. However, more calculations are needed than that of a larger ε. Note that x_k≠v_i is required by the ISODATA algorithm and equations (4)、(5). Consequently, the initial partition matrix must also satisfy the following conditions, in addition to the three basic ones of fuzzy clustering ^[4].

(1) R can’t be a constant matrix with equal elements;

(2) R can’t be a constant matrix with equal elements in a certain column or a certain row;

(3) If R is a single sample cluster, it must be separated in advance. Add it as a cluster after clustering is finished.

Thus, distortion of R selected by this way won’t happen during the iterative calculation.

2.4    Cluster validity

The above method gives a optimal partition corresponding to a certain partition number C, a partition matrix R, an error εand an exponential weight q. More local clusters can be obtained when initially selected values differ. So the partition coefficient F_c(R) and partition entropy H_c(R) are proposed to select the optimal cluster from all the local ones. They indicate the quality of the clustering solution.

2.4.1    Partition coefficient F(r)

If R∈M_c, then F_c(R)=1. That is, F_c(R) attains its extreme value for crisp partition. For this reason, the closer F_c(R) is to 1, the less fuzzy the result of clustering will be and so the better the clustering quality is.

2.4.2    Partition entropy

‘Entropy’ is a thermodynamics concept. Originally it was used to explain aspects of heat transfer. At a molecular level, it is a measure of the level of random behavior. It is also used as a measure of uncertainty in probability theory, giving a measure of the information volume left in information theory. Here, it was used as a measure of fuzziness (H_c). The closer the partition entropy is to zero, the more likely that a clustering effect is a valid representation of a data-set.

3    PRACTICAL EXAMPLE

3.1    Background ^[5]

Unit 2 of a certain hydropower station has a rated power of 23.4 MW at 375 rpm. During its preliminary operation, abnormal vibration at the main guide bearing was found within a load range of 40%~70%. The amplitude of vibration and displacement of shaft went up with increase of load. It came to a maximum when load reached 16.8 MW.

3.2    Selection of samples

The amplitude of vibration at each measuring point is selected as a sample of the working condition at 16.8 MW, when the unit vibrated most seriously. All the samples X={x_i| i=1,2, …, 5}, are listed in table 1.

3.3    Caculation of membership grade

Using equation (1), with a feature norm m=2, symptom membership grades are obtained as follows:

Consequently, corresponding fault membership grade values listed in table 2 are obtained based on equation [u_mn]=R [u_ln].  

3.4    Application of the dynamic isodata algorithm

The software can calculate different optimal partitions when ε、q and c are different for its structural design and ease of use.

Here, the number of faults N=16, number of feature norms m=2. The results are as follows when C=3, ε=0.0001, q=1.1 .

(1) Final partitions:  the first {v₃, v₈, v₁₄}

                 the second { v₄, v₅, v₉, v₁₁, v₁₃, v₁₆}

                 the third { v₁, v₂, v₆, v₇, v₁₀, v₁₂, v₁₅}

(2) Central vectors of each cluster:

                   V(1,1)=0.5320267      V(1,2)=0.3945668      

                   V(2,1)=0.3705752      V(2,2)=0.3577343

                   V(3,1)=0.1907992      V(3,2)=0.2463746      

(3) F_c(R)、H_c(R)

                   F_c(R)=0.9997085             H_c(R)=1.668719E-03

As mentioned above, the optimal final partition is only a relative one. It depends on the data initially chosen. Other results with different starting values are listed in table 3 for comparison.

3.5    Analysis of results

(1) The central vectors of the first cluster are the largest, compared with other clusters. So faults included in the first cluster are identified preferentially as reasons to cause the unit’s abnormal vibration. An explanation for each cluster vector may have no practical meaning. It is only a relative value. For instance, the vectors of the first cluster are the largest of the three. So we think the faults v₃, v₈ and v₁₄ included in the first cluster have the greatest probability of causing abnormal vibration. The second cluster and third clusters have progressively less probability of representing a meaningful association.

(2) Consequently, labyrinth clearance, main guide bearing and axis of the unit shaft must be checked first, to see whether they satisfy the specifications. The faults included in the second cluster might also be considered to ensure thorough fault diagnosis.

(3) The diagnosis has a high measure of reliability, since the partition coefficient is close to 1 and the partition entropy is close to 0.

3.6    Inspection of results

The site inspection record showed that a small clearance of the labyrinth ring is one of the vibration causes. This agrees with the fault diagnosis of ‘improper structure of seal ring and assembly clearance’ —v₁₄. Another reason is the ‘bent shaft’, which agrees with the diagnosis of ‘misalignment of shafts’—v₈ and ‘the bearing shaft and the bearing are not concentric, requiring a larger clearance between the bearing and shaft’—v₃.

4    CONCLUSION

This study shows that application of the fuzzy dynamic ISODATA algorithm, for vibration fault diagnosis of hydroelectric generating set, is feasible and has a degree of reliability. A fuzzy relationship matrix, corresponding symptom set and related fault set can be constructed based on each unit’s structure and design parameters. They are realized by changing the column and row of matrix. If it is used together with Expert Systems as the first step to categorize the optimal diagnosis range in condition monitoring and fault diagnosis, then the time taken for this work can be decreased greatly. From increased practical experience, the level of precision of the analysis is likely to be developed further.

Acknowledgements

This research is supported by the education ministry of china.

References

[1]    Wu Jinpei, Fuzzy diagnosis theory and application, Science and Technology Press, Beijing, 1995

[2]    Wen Xishen, Pattern recognition and condition monitoring, Defency and Technology University Press, Changsha, 1997 

[3]    Xiao Weishu, Fuzzy mathematics base and  its application, Aviation Industry Press, Beijing, 1992

[4]    Zhangyue, Fuzzy mathematics method and its application, Coal Industry Press, Beijing, 1992

[5]    Yao Dakun, Analysis for Self-excited Vibration of Francis-turbine, Technology of Large Electrical Machinery, 1998(5) 

                         Table 1    Preliminary vibration data    (mm)

                               Table 2    Fault membership grades

                              Table 3    Calculation results for different starting values